New Perspectives in Turbulence pp 55-103 | Cite as

# Rapid distortion theory as a means of exploring the structure of turbulence

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## Summary

Turbulence structure is discussed in terms of the different length scales of a turbulent flow—the large scale motions characteristic of the boundary conditions and forcing of the particular flow and, for high Reynolds numbers, the universal small scale motions. Evidence is presented that in shear flows and flows near boundaries the large scale structure of many different turbulent flows is similar. The analysis and understanding of different types and different regions of turbulent flows, and in particular their sensitivity to boundary and initial conditions, is clarified by using the classification of “rapidly changing turbulence” (RCT) and “slowly changing turbulence” (SCT), according to whether the time (*T* _{D}) over which fluid particles pass through (or near) changes in the mean flow or boundary conditions is much less or much greater than the characteristic time of the large scales of the turbulence (*T* _{L}). It is noted that in all unconfined turbulent flows, the turbulence structure adjusts so that, *T* _{D} ~ *T* _{L}, which implies that some features of the initial conditions, and boundary conditions can persist throughout the flow. A possible physical explanation is suggested.

The linear “Rapid distortion theory” (RDT) of turbulence and its assumptions are briefly reviewed here. It is shown that it is strictly applicable to the former category (RCT). But for some flows the linear theory has slowly changing solutions, or “statistical eigensolutions”, and these approximate to certain features of slowly changing turbulent flows (SCT). Examples are given of shear flows and turbulence near boundaries. In addition RDT can be used to simulate some aspects of the large eddy structures of turbulent shear flows.

The style of this paper is informal; for details of some of the results see Hunt & Carruthers (1990) (HC) and Carruthers, Fung, Hunt & Perkins (1990).

## Keywords

Shear Flow Turbulent Boundary Layer Isotropic Turbulence Vorticity Field Fluid Element## Preview

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## References

- Adrian, R.J. & Moin, P. 1988 Stochastic estimation of organized turbulent structure: homogeneous shear flow.
*J. Fluid Mech.***190**, 531–559.zbMATHCrossRefGoogle Scholar - Antonia, R.A. & Bisset, D.K. 1990 Spanwise structure in the wall region of a turbulent boundary layer.
*J. Fluid Mech.***210**, 437–458.CrossRefGoogle Scholar - Batchelor, G.K. 1953
*The Theory of Homogeneous Turbulence*. Cambridge University Press, 197pp.Google Scholar - Batchelor, G.K. 1955 The effective pressure, exerted by a gas in turbulent motion. In:
*Vistas in Astronomy*(ed. A. Beer), vol. 1, pp. 290–295, Pergamon.Google Scholar - Batchelor, G.K. 1967
*An Introduction to Fluid Dynamics*. Cambridge University Press, 615pp.Google Scholar - Batchelor, G.K. & Proudman, I. 1954 The effects of rapid distortion of a fluid in turbulent motion.
*Q.J. Mech. Appl. Math.***7**, 83.MathSciNetzbMATHCrossRefGoogle Scholar - Bavilagau, P.M. & Lykads, P.S. 1978 Turbulence memory in self-presering wakes.
*J. Fluid Mech.***89**, 589–606.CrossRefGoogle Scholar - Britter, R.E., Hunt, J.C.R. & Richards, K.J. 1981 Analysis and wind-tunnel studies of speed-up, roughness effects and turbulence over a two-dimensional hill.
*Q.J. Roy. Met. Soc.***107**, 91–110.CrossRefGoogle Scholar - Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation.
*J. Fluid Mech.***202**, 295–318.MathSciNetzbMATHCrossRefGoogle Scholar - Carruthers, D.J., Fung, J.C.H., Hunt, J.C.R. & Perkins R.J. 1989 The emergence of characteristic eddy motion in turbulent shear flows.
*Proc. Organized Structures and Turbulence in Fluid Mechanics*(ed. M. Lesieur & O. Metais), Kluwer Academic Publishers.Google Scholar - Carruthers, D.J. & Hunt, J.C.R. 1986 Velocity fluctuations near an interface between a turbulent region and a stably stratified layer.
*J. Fluid Mech.***165**, 475–501.zbMATHCrossRefGoogle Scholar - Carruthers, D.J. & Hunt, J.C.R. 1988 Turbulence, waves, and entrainment near density inversion layers. Proc. I.M.A. Conf. on “Stably Stratified Flow and Dense Gas Dispersion” (Ed. J.S. Puttock), Clarendon Press, pp. 77–96.Google Scholar
- Champagne, F.H., Harris, V.G. & Corrsin, S. 1970 Experiments on nearly homogeneous turbulent shear flow.
*J. Fluid Mech.***41**, 81–139.CrossRefGoogle Scholar - Comte-Bellot, G. & Corrsin, S. 1966 The use of a contraction to improve the isotropy of grid-generated turbulence.
*J. Fluid Mech.***25**, 657–682.CrossRefGoogle Scholar - Craik, A.D.D. & Criminale, W.O. 1986 Evolution of wavelike disturbances in shear flow: a class of exact solutions of the Navier-Stokes equations.
*Proc. R. Soc. Lond.***A406**, 13–26.MathSciNetGoogle Scholar - Craya, A. 1958 Contribution à l’analyse de la turbulence associée à des vitesses moyennes.
*P.S.T. Ministère de l’Air***345**.Google Scholar - Davidson, P.A., Hunt, J.C.R. & Moros, A. 1988 Turbulent recirculating flows in liquid metal magnetohydrodynamics.
*Prog. in Astronaut. Aeronaut.***111**, 400–420.Google Scholar - Deissler, R.G. 1968 Effects of combined two-dimensional shear and normal strain on weak locally homogeneous turbulence and heat transfer.
*J. Math. Phys.***47**, 320.Google Scholar - Domaradzki, J.A. & Rogallo, R.S. 1990 Local energy transfer and nonlocal interactions in homogeneous isotropic turbulence.
*Phys. Fluids***A2**, 413–426.Google Scholar - Dritschel, D. 1989 Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows.
*Comput. Phys. Rep.***10**, 77–146.CrossRefGoogle Scholar - Farge, M. & Holschneider, M. 1990 Wavelet analysis of coherent structures in two-dimensional flows. In
*Proc. IUTAM Symp. On Topological Fluid Mechanics*, pp. 765–776 (Ed. H.K. Moffatt & A. Tsinober), Cambridge University Press.Google Scholar - Ferré, J.A., Mumford, J.C., Savill, A.M. & Giralt, F 1990 Three-dimensional large-eddy motions and fine-scale activity in a plane turbulent wake.
*J. Fluid Mech.***210**, 371–414.CrossRefGoogle Scholar - Favre, A., Guitton, H., Lichnerowicz, A. & Wolff, E. 1988
*De la causalité à la finalité — a propos de la turbulence*, Maloine.Google Scholar - Fung, J.C.H. 1990 Kinematic Simulation of turbulent flow and particle motions. Ph.D dissertation University of Cambridge.Google Scholar
- Fung, J.C.H., Hunt, J.C.R., Malik, N.A. and Perkins, R.J. 1991 Kinematic simulation of homogeneous turbulent flows generated by unsteady random Fourier modes. Submitted to
*J. Fluid Mech.*Google Scholar - Gartshore, I.S., Durbin, P.A. & Hunt, J.C.R. 1983 The production of turbulent stress in a shear flow by irrotational fluctuations.
*J. Fluid Mech.***137**, 307–329.CrossRefGoogle Scholar - Gaster, M., Kit, E. & Wygnanski, I. 1985 Large-scale structures in a forced turbulent mixing layer.
*J. Fluid Mech.***150**, 23–39.CrossRefGoogle Scholar - Gilbert, A. 1988 Spiral structures and spectra in two-dimensional turbulence.
*J. Fluid Mech.***193**, 475–497.MathSciNetzbMATHCrossRefGoogle Scholar - Goldstein, M.E. 1978 Unsteady vortical and entropic distortion of potential flow round arbitrary obstacles.
*J. Fluid Mech.***89**, 433–468.zbMATHCrossRefGoogle Scholar - Goldstein, M.E. & Durbin, P.A. 1980 The effect of finite turbulence spatial scale on the amplification of turbulence by a contracting stream.
*J. Fluid Mech.***98**, 473–508.zbMATHCrossRefGoogle Scholar - Hayakawa, M. & Hussain, F. 1989 Three dimensionality of organized structures in a plane turbulent wake.
*J. Fluid Mech.***206**, 375–404.CrossRefGoogle Scholar - Ho, C.M. & Huerre, P. 1984 Perturbed free shear layers.
*Ann. Rev. Fluid Mech.***16**, 365–424.CrossRefGoogle Scholar - Hunt, J.C.R. 1973 A theory of turbulent flow round two-dimensional bluff bodes.
*J. Fluid Mech.***61**, 625–706.MathSciNetzbMATHCrossRefGoogle Scholar - Hunt, J.C.R. 1978 A review of the theory of rapidly distorted turbulent flow and its applications.
*Proc. of XIII Biennial Fluid Dynamics symp., Kortowo, Poland, Fluid Dyn. Trans.***9**, 121–152.Google Scholar - Hunt, J.C.R. 1984 Turbulence structure in thermal convection and shear-free boundary layers.
*J. Fluid Mech.***138**, 161–184.zbMATHCrossRefGoogle Scholar - Hunt, J.C.R. 1987 Vorticity and vortex dynamics in complex turbulence flow.
*Trans. Can. Soc. Mech. Eng.***11**, 21–35.Google Scholar - Hunt, J.C.R. 1988 Studying turbulence using direct numerical simulation: 1987 Center for Turbulence Research NASA Ames-Stanford Summer Programme.
*J. Fluid Mech.***190**, 375–392.zbMATHCrossRefGoogle Scholar - Hunt, J.C.R. 1990 Review of
*De la causalité à la finalite — a propos de la turbulence*, Maloine, by Favre, A., Guitton, H., Lichnerowicz, A. & Wolff, E., 1988. Euro. J. of Mech. B/Fluids.Google Scholar - Hunt, J.C.R. & Graham, J.M.R. 1978 Freestream turbulence near plane boundaries.
*J. Fluid Mech.***84**, 209–235.MathSciNetzbMATHCrossRefGoogle Scholar - Hunt, J.C.R., Kaimal, J.C. & Gaynor, E. 1988 Eddy structure in the convective boundary layer — new measurements and new concepts.
*Q. J. Roy. Met. Soc.***114**, 827–858.Google Scholar - Hunt, J.C.R., Moin, P., Lee, M., Moser, R.D., Spalart, P. & Mansour, N.N. 1989 Cross correlation and length scales in turbulent flow near surfaces. In
*Proc. of the Second European Turbulence Conference*. Advances in Turbulence 2 (ed. H.H. Fernholz & H.E. Fiedler), pp. 128–134, Springer-Verlag.Google Scholar - Hunt, J.C.R., Stretch, D.D. & Britter R.E. 1988 Length scales in stably stratified turbulent flows and their use in turbulence models. In
*Proc. IMA Conf. on Stably Stratified Flow and Dense Gas Dispersion*(ed. J.S. Puttock), Chester, England, April 1986, pp. 285–321, Clarendon.Google Scholar - Hunt, J.C.R. & Carruthers, D.J. 1990 Rapid distortion theory and the ‘problem’ of turbulence.
*J. Fluid Mech.***212**, 497–532.MathSciNetzbMATHCrossRefGoogle Scholar - Hunt, J.C.R. & Vassilicos, J.C. 1991 Kolmogrov’s contributions to the physical and geometrical understanding of turbulent flows and recent developments.
*Proc. Roy. Soc. A*, July 1991.Google Scholar - Hussain, A.K.M.F. 1986 Coherent structures and turbulence.
*J. Fluid Mech.***173**, 303–356.CrossRefGoogle Scholar - Husain, H.S. & Hussain, F. 1990 Subharmonic resonance in a shear layer. In
*Proc. of the Second European Turbulence Conference*. Advances in Turbulence 2 (ed. H.H. Fernholz & H.E. Fiedler), pp. 96–101, Springer-Verlag.Google Scholar - Jeandel, D., Brison, J.F. & Mathieu, J. 1978 Modelling methods in physical and spectral space.
*Phys. Fluids***21**, 169–182.MathSciNetzbMATHCrossRefGoogle Scholar - Kida, S. & Hunt, J.C.R. 1989 Interaction between turbulence of different scales over short times.
*J. Fluid Mech.***201**, 411–445.MathSciNetzbMATHCrossRefGoogle Scholar - Klebanoff, P.S. & Sargent, L.H. 1962 The three-dimensional nature of boundary layer.
*J. Fluid Mech.***12**, 1–34.zbMATHCrossRefGoogle Scholar - Kellogg, R.M. & Corrsin, S. 1980 Evolution of a spectrally local disturbance in grid-generated, nearly isotropic turbulence.
*J. Fluid Mech.***96**, 641–669.CrossRefGoogle Scholar - Kline, S.J. 1981 Universal or zonal modelling — the road ahead. In
*Proc. of Complex Turbulent flows*. Stanford University, California. vol. II, pp. 991–998.Google Scholar - Kline, S.J., Reynolds, W.C., Schraub, F.A. & Runstadler, P.W. 1967 The structure of turbulent boundary layers.
*J. Fluid Mech.***30**, 741–773CrossRefGoogle Scholar - Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.
*Acad. Sci., USSR***30**, 301–305.Google Scholar - Komori, S., Ueda, H., Ogino, F. & Mizushina, T. 1983 Turbulence structure in stably stratified open channel flow.
*J. Fluid Mech.***130**, 13–26.CrossRefGoogle Scholar - Landahl, M.T. 1990 On sublayer streaks. To appear in J. Fluid Mech.Google Scholar
- Launder, B.E., Reece, G.J. & Rodi, W. 1975 Progress in the development of a Reynolds stress turbulence closure.
*J. Fluid Mech.***68**, 537–566.zbMATHCrossRefGoogle Scholar - Launder, B.E. & Spalding, D.B. 1972
*Mathematical Models of turbulence*. Academic.Google Scholar - Lee, M.J. & Hunt, J.C.R. 1988 The structure of sheared turbulence near a boundary.
*Report, Center for Turbulence Research, Stanford, No. CTR-S88*, pp. 221–242.Google Scholar - Lee, M.J. & Hunt, J.C.R. 1989 The structure of shear turbulence near a plane boundary. In
*Proc. of the Seventh Sypm. on Turbulent Shear Flows*. Stanford University. pp. 8.1.1–8.1.6.Google Scholar - Lee, M.J. Kim, J. & Moin, P. 1987 Turbulent structure at high shear rate. In:
*Sixth Symposium on Turbulent Shear Flows, Toulouse, France*(ed. F. Durst*et al.*), pp. 22.6.1–22.6.6.Google Scholar - Lesieur, M. 1987
*Turbulence in Fluids*. Martinus Nijhoff.Google Scholar - Lesieur, M & Metais, O. 1989
*Proc. Pole European Pilote de Turbulence (PEPIT), ERCOFTAC Summer School, Lyon July 1989*.Google Scholar - Leslie, D.C. 1973
*Developments in the Theory of Turbulence*. Clarendon.Google Scholar - Liu, J.T.C. 1989 Contributions to the understanding of large-scale coherent structures in developing free turbulent shear flows.
*Adv. in Appl. Mech.***26**, 535–540.Google Scholar - Lumley, J. 1965 The structure of inhomogeneous turbulent flows.
*Proc. Int. Coll. on Radio Wave Propagation*. (ed. A.M. Yaglom & V.I. Takasky).*Dokl. Akad. Nauk. SSSR.*166–178.Google Scholar - Lumley, J.L. 1978 Computational modelling of turbulent flows.
*Adv. in Appl. Mech.***18**, 126–176.Google Scholar - Malkus, W.V.R. 1956 Outline of a theory of turbulent shear flow.
*J. Fluid Mech.***1**, 521–539.MathSciNetzbMATHCrossRefGoogle Scholar - Mason, P.J. & Sykes R.I. 1982 A two-dimensional numerical study of horizontal roll vortices in an inversion capped planetary boundary layer.
*Q. J. Roy. Met. Soc.***108**, 801–823.CrossRefGoogle Scholar - Maxey, M.R. 1978 Aspects of Unsteady Turbulent Shear Flow. Ph.D. Dissertation, University of Cambridge.Google Scholar
- Maxey, M.R. 1982 Distortion of turbulence in flows with parallel streamlines.
*J. Fluid Mech.***124**, 261–282.zbMATHCrossRefGoogle Scholar - Melander M.V. & Hussain, F. 1990 Cut-and connect of antoparaellel vortex tubes. In
*Proc. IUTAM Symp. On Topological Fluid Mechanics*. Cambridge University Press.Google Scholar - Moffatt, H.K. 1967 On the suppression of turbulence by a uniform magnetic field.
*J. Fluid Mech.***28**, 571–592.CrossRefGoogle Scholar - Moffatt, H.K. 1984 Simple topological aspects of turbulent vorticity dynamics. In:
*Turbulence and Chaotic Phenomena in Fluids*(ed. T. Tatsumi), pp. 223–230, Elsevier.Google Scholar - Monin, A.S. & Yaglom, A.M. 1971
*Statistical Theory of Turbulence*, vol. II. MIT Press.Google Scholar - Mumford, J.C. 1982 The structure of the large eddies in fully developed turbulent shear flows. Part 1. The plane jet.
*J. Fluid Mech.***118**, 241–268.CrossRefGoogle Scholar - Murakami, S. & Mochida, A. 1988 3D numerical simulation of airflow around a cubic model by means of a
*k*— ɛ model.*J. Wind. Eng. Indust. Aerodyn.***31**, No. 2.Google Scholar - Naish, A. & Smith, F.T. 1988 The turbulent boundary layer and wake of an aligned flat plate.
*J. Eng. Sci. Maths.***22**, 15–42.CrossRefGoogle Scholar - Phillips, O.M. 1955 The irrotational motion outside a free turbulent boundary.
*Proc. Camb. Phi. Soc.***51**, 220–229.zbMATHCrossRefGoogle Scholar - Reynolds, W.C. 1989 Effects of rotation on homogeneous turbulence. In
*Proc. of Australasian Conf. on Fluid Mechanics*.Google Scholar - Rogallo, R.S. 1981 Numerical experiments in homogeneous turbulence.
*NASA Tech. Memo.*81315.Google Scholar - Rogers, M.M. 1991 The structure of a passive scalar field with a uniform mean scalar gradient in rapidly sheared homogeneous turbulent flow.
*Phys. Fluids***A3**, 144–154.Google Scholar - Rogers, M.M. & Moin, P. 1987 The structure of the vorticity field in homogeneous turbulent flows.
*J. Fluid Mech.***176**, 33–66.CrossRefGoogle Scholar - Sabot, J. & Comte-Bellot 1976 Intermittency of coherent structures in the core region of fully develop turbulent pipe flow.
*J. Fluid Mech.***74**, 767–796.CrossRefGoogle Scholar - Savill, A.M. 1987 Recent developments in rapid-distorion theory.
*Ann. Rev. Fluid Mech.***19**, 531–570.CrossRefGoogle Scholar - Sreenivasan, K.R. 1985 The effect of a contraction on a homogeneous turbulent shear flow.
*J. Fluid Mech.***154**, 187–213.CrossRefGoogle Scholar - Sreenivasan, K.R. & Narasimha 1978 Rapid distortion of axisymmetric turbulence.
*J. Fluid Mech.***84**, 497–516.zbMATHCrossRefGoogle Scholar - Sulem, P.L., She, Z.S., Scholl, H. & Frisch, U. 1989 Generation of large-scale structures in three dimensional flow lacking partity-invariance.
*J. Fluid Mech.***205**, 341–358.MathSciNetCrossRefGoogle Scholar - Tennekes, H. & Lumley, J.L. 1971
*A First Course in Turbulence*. MIT Press.Google Scholar - Townsend, A.A. 1970 Entrainment and the structure of turbulent flow.
*J. Fluid Mech.***41**, 13–46.zbMATHCrossRefGoogle Scholar - Townsend, A.A. 1976
*Structure of turbulent Shear Flow*. Cambridge University Press.Google Scholar - Townsend, A.A. 1980 The response of sheared turbulence to additional distortion.
*J. Fluid Mech.***98**, 171–191.zbMATHCrossRefGoogle Scholar - van Haren, L. 1991 1991 ERCOFTAC Summer School in Rutherford Appleton Laboratory, U.K.Google Scholar
- Weber, W. 1868 Über eine Transformation der hydrodynamischen Gleichungen.
*J. Reine Angew. Math.***68**, 286.zbMATHCrossRefGoogle Scholar - Wong, H.Y.W. 1985 Shear-free turbulence and secondary flow near angled and curved surfaces. Ph.D dissertation University of Cambridge.Google Scholar
- Wray, A. & Hunt, J.C.R. 1990 Algorithms for classification of turbulent structures. In
*Proc. IUTAM Symp. On Topological Fluid Mechanics*, pp. 95–104 (Ed. H.K. Moffatt & A. Tsinober), Cambridge University Press.Google Scholar - Wyngaard, J.C. & Cote, O.R. 1972 Modelling the buoyancy driven mixed layer.
*J. Atmos. Sci.***33**, 1974–1988.Google Scholar - Yakhot, V. & Orszag, S.A. 1986 Renormalization group analysis of turbulence. I. Basic theory.
*J. Sci. Comput.***1**, 3.MathSciNetzbMATHCrossRefGoogle Scholar - Zhou, M.D. 1989 A new modelling approach to complex turbulent shear flow. In
*Proc. of the Second European Turbulence Conference*. Advances in Turbulence 2 (ed. H.H. Fernholz & H.E. Fiedler), pp. 146–150, Springer-Verlag.Google Scholar